A new phase transition in the parabolic Anderson model with partially duplicated potential

Muirhead, S. and Pymar, Richard and Sidorova, N. (2019) A new phase transition in the parabolic Anderson model with partially duplicated potential. Stochastic Processes and their Applications 129 (11), pp. 4704-4746. ISSN 0304-4149.

Abstract

We investigate a variant of the parabolic Anderson model, introduced in previous work, in which an i.i.d. potential is partially duplicated in a symmetric way about the origin, with each potential value duplicated independently with a certain probability. In previous work we established a phase transition for this model on the integers in the case of Pareto distributed potential with parameter α >1 and fixed duplication probability p∈(0,1): if α ≥2 the model completely localises, whereas if α ∈(1,2) the model may localise on two sites. In this paper we prove a new phase transition in the case that α ≥2 is fixed but the duplication probability p(n) varies with the distance from the origin. We identify a critical scale p(n)→1, depending on α, below which the model completely localises and above which the model localises on exactly two sites. We further establish the behaviour of the model in the critical regime.

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